xrge0iifiso - Mathbox for Thierry Arnoux

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Theorem xrge0iifiso 34567

Description: The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)

**Hypothesis**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))

**Assertion**
Ref	Expression
xrge0iifiso	⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Distinct variable group: 𝑥,𝐹

Proof of Theorem xrge0iifiso Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 13561	. . 3 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 13270	. . 3 ⊢ < Or ℝ^*
3		soss 5578	. . 3 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. 2 ⊢ < Or (0[,]1)
5		iccssxr 13561	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
6		cnvso 6291	. . . . 5 ⊢ ( < Or ℝ^* ↔ ^◡ < Or ℝ^*)
7	2, 6	mpbi 233	. . . 4 ⊢ ^◡ < Or ℝ^*
8		sopo 5577	. . . 4 ⊢ (^◡ < Or ℝ^* → ^◡ < Po ℝ^*)
9	7, 8	ax-mp 5	. . 3 ⊢ ^◡ < Po ℝ^*
10		poss 5560	. . 3 ⊢ ((0[,]+∞) ⊆ ℝ^* → (^◡ < Po ℝ^* → ^◡ < Po (0[,]+∞)))
11	5, 9, 10	mp2 9	. 2 ⊢ ^◡ < Po (0[,]+∞)
12		xrge0iifhmeo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
13	12	xrge0iifcnv 34565	. . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 = +∞, 0, (exp‘ -𝑧))))
14	13	simpli 489	. . 3 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
15		f1ofo 6832	. . 3 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)–onto→(0[,]+∞))
16	14, 15	ax-mp 5	. 2 ⊢ 𝐹:(0[,]1)–onto→(0[,]+∞)
17		0xr 11356	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
18		1xr 11368	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
19		0le1 11839	. . . . . . . 8 ⊢ 0 ≤ 1
20		snunioc 13611	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
21	17, 18, 19, 20	mp3an 1490	. . . . . . 7 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
22	21	eleq2i 2852	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ 𝑤 ∈ (0[,]1))
23		elun 4152	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
24	22, 23	bitr3i 280	. . . . 5 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
25		velsn 4599	. . . . . . 7 ⊢ (𝑤 ∈ {0} ↔ 𝑤 = 0)
26		elunitrn 13598	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
27	26	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ ℝ)
28		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 0 < 𝑧)
29		elicc01 13597	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
30	29	simp3bi 1165	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ≤ 1)
31	30	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ≤ 1)
32		1re 11308	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		elioc2 13540	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1)))
34	17, 32, 33	mp2an 705	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1))
35	27, 28, 31, 34	syl3anbrc 1362	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ (0(,]1))
36		pnfxr 11363	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
37		0le0 12444	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
38		ltpnf 13249	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℝ → 1 < +∞)
39	32, 38	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 1 < +∞
40		iocssioo 13570	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
41	17, 36, 37, 39, 40	mp4an 706	. . . . . . . . . . . . . 14 ⊢ (0(,]1) ⊆ (0(,)+∞)
42		ioorp 13556	. . . . . . . . . . . . . 14 ⊢ (0(,)+∞) = ℝ⁺
43	41, 42	sseqtri 3956	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ ℝ⁺
44	43	sseli 3904	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0(,]1) → 𝑧 ∈ ℝ⁺)
45		relogcl 26903	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → (log‘𝑧) ∈ ℝ)
46	45	renegcld 11743	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) ∈ ℝ)
47		ltpnf 13249	. . . . . . . . . . . . . 14 ⊢ ( -(log‘𝑧) ∈ ℝ → -(log‘𝑧) < +∞)
48	46, 47	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) < +∞)
49		brcnvg 5856	. . . . . . . . . . . . . 14 ⊢ ((+∞ ∈ ℝ^* ∧ -(log‘𝑧) ∈ ℝ) → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
50	36, 46, 49	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
51	48, 50	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → +∞^◡ < -(log‘𝑧))
52	44, 51	syl 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < -(log‘𝑧))
53	12	xrge0iifcv 34566	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → (𝐹‘𝑧) = -(log‘𝑧))
54	52, 53	breqtrrd 5132	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < (𝐹‘𝑧))
55	35, 54	syl 18	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → +∞^◡ < (𝐹‘𝑧))
56	55	ex 418	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) → (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧)))
57		breq1 5105	. . . . . . . . 9 ⊢ (𝑤 = 0 → (𝑤 < 𝑧 ↔ 0 < 𝑧))
58		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = (𝐹‘0))
59		0elunit 13600	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
60		iftrue 4487	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
61		pnfex 11362	. . . . . . . . . . . . 13 ⊢ +∞ ∈ V
62	60, 12, 61	fvmpt 6993	. . . . . . . . . . . 12 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
63	59, 62	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹‘0) = +∞
64	58, 63	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = +∞)
65	64	breq1d 5112	. . . . . . . . 9 ⊢ (𝑤 = 0 → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ +∞^◡ < (𝐹‘𝑧)))
66	57, 65	imbi12d 347	. . . . . . . 8 ⊢ (𝑤 = 0 → ((𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)) ↔ (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧))))
67	56, 66	imbitrrid 249	. . . . . . 7 ⊢ (𝑤 = 0 → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
68	25, 67	sylbi 220	. . . . . 6 ⊢ (𝑤 ∈ {0} → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
69		simpll 779	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ (0(,]1))
70	26	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ ℝ)
71		0re 11310	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 ∈ ℝ)
73	43	sseli 3904	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ⁺)
74	73	rpred 13164	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ)
75	74	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ ℝ)
76		elioc2 13540	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1)))
77	17, 32, 76	mp2an 705	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1))
78	77	simp2bi 1164	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 0 < 𝑤)
79	78	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑤)
80		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 < 𝑧)
81	72, 75, 70, 79, 80	lttrd 11471	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑧)
82	30	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ≤ 1)
83	70, 81, 82, 34	syl3anbrc 1362	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ (0(,]1))
84	69, 83	jca 521	. . . . . . . 8 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)))
85	73	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑤 ∈ ℝ⁺)
86	85	relogcld 26951	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑤) ∈ ℝ)
87	44	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑧 ∈ ℝ⁺)
88	87	relogcld 26951	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑧) ∈ ℝ)
89	86, 88	ltnegd 11894	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((log‘𝑤) < (log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
90		logltb 26928	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
91	73, 44, 90	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
92		negex 11555	. . . . . . . . . . . . 13 ⊢ -(log‘𝑤) ∈ V
93		negex 11555	. . . . . . . . . . . . 13 ⊢ -(log‘𝑧) ∈ V
94	92, 93	brcnv 5859	. . . . . . . . . . . 12 ⊢ ( -(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤))
95	94	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ( -(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
96	89, 91, 95	3bitr4d 314	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
97	96	biimpd 232	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → -(log‘𝑤)^◡ < -(log‘𝑧)))
98	12	xrge0iifcv 34566	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0(,]1) → (𝐹‘𝑤) = -(log‘𝑤))
99	98, 53	breqan12d 5118	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
100	97, 99	sylibrd 262	. . . . . . . 8 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
101	84, 80, 100	sylc 66	. . . . . . 7 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
102	101	exp31 425	. . . . . 6 ⊢ (𝑤 ∈ (0(,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
103	68, 102	jaoi 871	. . . . 5 ⊢ ((𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
104	24, 103	sylbi 220	. . . 4 ⊢ (𝑤 ∈ (0[,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
105	104	imp 412	. . 3 ⊢ ((𝑤 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
106	105	rgen2 3202	. 2 ⊢ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
107		soisoi 7336	. 2 ⊢ ((( < Or (0[,]1) ∧ ^◡ < Po (0[,]+∞)) ∧ (𝐹:(0[,]1)–onto→(0[,]+∞) ∧ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))) → 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞)))
108	4, 11, 16, 106, 107	mp4an 706	1 ⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ⊆ wss 3895 ∪ cun 4149 ifcif 4481 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 Po wpo 5556 Or wor 5557 ^◡ccnv 5649 –onto→wfo 6536 –1-1-onto→wf1o 6537 ‘cfv 6538 Isom wiso 6539 (class class class)co 7420 ℝcr 11199 0cc0 11200 1c1 11201 +∞cpnf 11340 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 -cneg 11542 ℝ⁺crp 13120 (,)cioo 13476 (,]cioc 13477 [,]cicc 13479 expce 16227 logclog 26882

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884

This theorem is used by: xrge0iifhmeo 34568

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